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Depends on which area of pure mathematics.

One of my strongest mathematical interests is graph theory, in part because networks are incredibly pure abstract mathematical objects which you can draw lots of conclusions about on a purely logical basis, and in part because they can be used to model so many real-world phenomena. As a result, even modest propositions in that particular area have lots of real-world consequences.

History also strongly suggests that even the most historically useless pure maths can have tremendously important applications and consequences further down the line, some choice examples being radon transformations, modular arithmetic and eigenspace. It would be an incredibly bold statement to say a particular area of pure maths is completely without real-world consequence. There's an awful lot of remaining time for even the most esoteric theorem to be put to use.

__Emil__14y00

Apologies in advance for nitpicking, but the heuristic is to ask what are the real-world consequences of propositions in this discipline being right or wrong, not whether the discipline has real-world consequences. So what are the propositions of mathematics that can be right or wrong? Clearly a published theorem can be right or wrong, but most are correct. What can be right or wrong is what areas of pure mathematics people consider to be interesting. I would say that these propositions can be right or wrong, and do have "real world" consequences. People used to think graph theory was not interesting - they were wrong.

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18 Which Fields Are Underserved?

by [anonymous]

15th Sep 2011

1 min read

18

I'm currently about 2/3rds through Jane Jacobs' "The Death and Life of Great American Cities". This is one of the defining works of modern urban planning, and Jane Jacobs is considered one of our most important urban planning thinkers.

It's a fine book, but I found myself surprised at just how unimpressive it is for a work of such supposed importance. Her hypothesis is simple enough that I can summarize it here:

-Cities succeed by having many people using the city streets throughout the day. Large numbers of people keep the streets safe, and provide enough traffic for businesses to thrive.

-To achieve this constant stream of people, streets should have a large variety of different businesses which are utilized at different times, and should eliminate barriers that prevent the flow of people.

Jacobs' reaches this hypothesis through her own observations of various cities, along with a few bits of data concerning densities, crime rates, etc. But there's no systematic examination of data, no meticulously constructed arguments, and no addressing of criticisms or alternate explanations. Evidently, all that it takes to be a great work in urban planning is the barest rudiments of basic science. (This isn't the first time I've been critical of supposed great works in this field.)

It strikes me that, for whatever reason, Urban Planning is an underserved field* - the scholarship behind it doesn't compare to, say, the quality of work done in evolution, or cognitive psychology.

I have my theories for why this might be**, but it got me thinking - which other fields show a distinct lack of quality work done in them? What other fields are underserved?

*There is of course the possibility that I'm unfamiliar with more recent, higher quality urban planning literature.

**Namely, that urban planning is an offshoot of architecture, which has tended to value on aesthetic judgement and intuition over empiricism and rigorously constructed arguments.

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[-]sixes_and_sevens14y110

Any field that attempts to analyse real-world phenomena as if they were a piece of literature. That's a bloody good start.

I've wanted to make some sort of post to this effect myself, but (ironically) couldn't come up with a coherent theme to draw all the ideas together.

I'm currently working my way through undergrad economics, and regularly notice people expounding upon their home-brew economic theories that wouldn't fly, make no sense, or have well-established theory or evidence opposing them. When I respond "this probably wouldn't work because of x", the most frustrating response is a wholesale rejection of economics as a legitimate field with useful findings. They don't engage with the economic arguments because they don't see the point in establishing the basic framework. This happens with alarming frequency.

The trouble is that I have a blacklist of fields that I basically don't think are worth my time to study because they look like spurious nonsense. On what basis is it reasonable for me to dismiss, say, a feminist post-structuralist discourse analysis of the recent banking crisis without bothering to engage with its arguments, and simultaneously criticise someone else for being wilfully ignorant of my own favoured disciplines?

My current rule of thumb, which largely seems to work, is to ask "what are the real-world consequences of propositions in this discipline being right or wrong?" It obviously doesn't distinguish all spurious nonsense from all useful disciplines, (the real-world consequences of homeopathy being right are enormous; it just happens to be conclusively wrong), but it does highlight which fields of study are getting work done, information-theoretically speaking, and which are sinkholes for effort without producing any practicable information.

[-][anonymous]14y40

One of the reasons I'm picking on urban planning here is that it seems like the consequences of it are enormous, given the importance of cities as generators of growth and innovation. (Though it's possible there's not in fact much difference between "successful" cities and "unsuccessful" ones.)

[-]jhuffman14y00

Urban planning seems similar to economics in some important respects. One of these, is that in practice the field is used as a garden of many different theories and tools which are selected from as needed to ex post facto justify political positions that are genuinely supported by unstated biases. Whatever crazy idea you have, you can be sure somewhere someone is receiving public or private funding to try and make it look legitimate.

[-]benelliott14y30

My current rule of thumb, which largely seems to work, is to ask "what are the real-world consequences of propositions in this discipline being right or wrong?"

Interesting heuristic. I'd be intrigued to hear what it says about pure mathematics?

[-]sixes_and_sevens14y30

Depends on which area of pure mathematics.

[-]__Emil__14y00

[-]sixes_and_sevens14y00

I'm not sure interestingness is really the focus of the issue. I'm sure feminist post-structuralist discourse analyses of the recent banking crisis are very interesting to people interested in the subject, but I still don't think it has any power to deduce true facts about the universe.

I do have another heuristic which is a little less straightforward to apply, but a little more selective: would a society of humans kept isolated from our own for thousands of years develop a similar discipline with the same essential elements as our version? Pure maths definitely passes that one, while something like Jungian analysis probably wouldn't.

[-]asr14y00

There's a lot of places where we'd have catastrophic consequences if we had wrong beliefs about pure mathematics. For example, public-key crypto would fall apart if mathematicians were severely mistaken about finite groups and their relationship to prime numbers. And we couldn't be very wrong about real analysis before we'd notice something the matter with calculus.

I would have said that math is a degenerate case for such a heuristic because we so seldom are wrong about it.

[-]torekp14y00

I wouldn't say that economics is an illegitimate field without useful findings, but it may well be underserved. It contains a lot of elegant mathematical superstructure build on shaky foundations. Treating normative theories like Von Neumann-Morgenstern utility as if they were descriptive is of very limited ... utility. As Daniel Hausman, cited by Leiter, wrote,

[T]he justification for a particular paradigm or research program, like the justification for the commitment to economics as a separate science, is success and progress, including especially empirical success and progress. Since economics has not been very successful and has not made much empirical progress, economists should be exploring alternatives. . ..[U]nless equilibrium theory [the core of what makes economics a distinct science according to Hausman] has captured the major causes of economic phenomena, the separate science of economics can never be successful. If, as seems likely to me, there are systematic failings of human rationality, and economic behavior is significantly influenced by many motive forces, apart from consumerism and diminishing marginal rates of substitution, then equilibrium theory is not a very good theory, whether or not there is anything better. ...

Many (behavioral, especially) economists have proposed partial models accounting for a given irrationality or deviance from standard choice theory. What I haven't heard of - and admittedly I don't follow the subject closely - is any synthesis that covers a wide range of real human choice patterns.

[-]FiftyTwo14y60

The seeming rigorousness of a field probably correlates with how easy it is to make repeatable experiments. For physics and chemistry it's extremely easy; animal biology and psychology a bit less so; and for political science, economics and urban planning, near impossible.

[-]Morendil14y60

Many things that are obvious in retrospect are anything but at the time the important debates happen, and they become obvious in retrospect precisely because someone has come along who paid attention to what - in retrospect - really mattered.

See James Scott's Seeing Like a State (which puts Jacobs' work in that kind of broader perspective, largely in line with calcsam's comment) and Duncan Watt's Everything is Obvious (which lays out the research on why the social sciences are especially prone to that phenomenon).

[-][anonymous]14y30

Right, inferential gaps always look smaller when they're behind you.

I'm not claiming that her insights were obvious, just that they weren't especially well constructed or well supported and that (critically) that doesn't seem to have improved with time.

[-]calcsam14y60

Consider Le Corbusier, Robert Moses, etc. These men combined methods which claimed to be scientific. Corbusier tried to maximize population density; Moses, to maximize road construction.

But they were working in very intricate, complicated systems and ignored the effects that maximizing their favorite metric would have on everything else. They dictated everything from the center and ignored local knowledge.

This is what we call dangerous knowledge.

The failure of these methods -- "the projects" housing inspired by Corbusier, Moses's neighborhood destruction, helped trigger -- as far as I understand -- the current focus on aesthetics and intuition. It's a reaction to that, a "risk-averse" strategy to picking the wrong metrics and trying to maximize/minimize them.

A parallel example might be Robert McNamara and the whiz kids turning into the Best and the Brightest in Vietnam.

[-][anonymous]14y30

I don't really see a focus on aesthetics and intution as a "new" focus, or something that was turned to as a reaction to previous urban planning. I haven't read the previous works, but what seems to set Jacobs apart is that she didn't merely base her judgments on what was aesthetically pleasing - that she actually went out and did basic science, collecting observations and forming a hypothesis that explained them.

[-]Douglas_Knight14y20

Did Le Corbusier really try to maximize density? Because he utterly failed. While the original post complains about how little Jacobs measured, it was a lot more than Le Corbusier.

[-]Hyena14y10

Corbusier, for his part, was actually part of the fashion for science which gains speed in the 19th century. It's about the "aesthetic" of science more than actual science, like how science is depicted in a comic book (for example).

[-]Hyena14y50

The major problem for urban planning is that it's expensive and interdisciplinary; I'd argue that one of the best works in "urban planning" is actually "The High Cost of Free Parking", a book about economics. A lot of the research, however, is likely to fall under civil engineering, as it often deals with traffic planning and environmental impacts of infrastructure.

[-]Matt_Simpson14y50

As a side note to your main point, I vaguely remember some economists collecting some data and putting Jacobs' theories to the test - and they seemed to check out. A quick google search turned up nothing, but I didn't look for long.

[-]Tyrrell_McAllister14y120

Are you thinking of Luis Bettencourt and Geoffrey West? They're physicists, but your description reminded me of these paragraphs:

It’s when West switches the conversation from infrastructure to people that he brings up the work of Jane Jacobs, the urban activist and author of “The Death and Life of Great American Cities.” Jacobs was a fierce advocate for the preservation of small-scale neighborhoods, like Greenwich Village and the North End in Boston. The value of such urban areas, she said, is that they facilitate the free flow of information between city dwellers. To illustrate her point, Jacobs described her local stretch of Hudson Street in the Village. She compared the crowded sidewalk to a spontaneous “ballet,” filled with people from different walks of life. School kids on the stoops, gossiping homemakers, “business lunchers” on their way back to the office. While urban planners had long derided such neighborhoods for their inefficiencies — that’s why Robert Moses, the “master builder” of New York, wanted to build an eight-lane elevated highway through SoHo and the Village — Jacobs insisted that these casual exchanges were essential. She saw the city not as a mass of buildings but rather as a vessel of empty spaces, in which people interacted with other people. The city wasn’t a skyline — it was a dance.

If West’s basic idea was familiar, however, the evidence he provided for it was anything but. The challenge for Bettencourt and West was finding a way to quantify urban interactions. As usual, they began with reams of statistics. The first data set they analyzed was on the economic productivity of American cities, and it quickly became clear that their working hypothesis — like elephants, cities become more efficient as they get bigger — was profoundly incomplete. According to the data, whenever a city doubles in size, every measure of economic activity, from construction spending to the amount of bank deposits, increases by approximately 15 percent per capita. It doesn’t matter how big the city is; the law remains the same. “This remarkable equation is why people move to the big city,” West says. “Because you can take the same person, and if you just move them to a city that’s twice as big, then all of a sudden they’ll do 15 percent more of everything that we can measure.” While Jacobs could only speculate on the value of our urban interactions, West insists that he has found a way to “scientifically confirm” her conjectures. “One of my favorite compliments is when people come up to me and say, ‘You have done what Jane Jacobs would have done, if only she could do mathematics,’ ” West says. “What the data clearly shows, and what she was clever enough to anticipate, is that when people come together, they become much more productive.”

[-]Matt_Simpson14y20

That it was economists is what I'm most certain of about my memory, but this is interesting too!

[-]gwern14y30

Would there be any point in trying to make urban planning better? I'm thinking here of a Scott Aaronson quote.

[-][anonymous]14y20

I can't think of any particular reason why urban planning need to suck - the problems with it seem to be based on historical happenstance, not structural necessity.

That's interesting though - are there any fields are that suck purely out of necessity? What's the mechanism that causes it?

[-]michael6114y20

I've worked in the field of urban construction for 45 yrs or so, and I think qwern's point is well taken. Urban planning meetings are complex affairs involving many competing interests. To expect a group of humans, with differing agendas to always make rational decisions is not going to happen in the near term. Until something is worked out to improve human thinking and decision making we'll have to keep muddling along. Having worked with it I am amazed we do as well as we do.

[-]gwern14y10

That's interesting though - are there any fields are that suck purely out of necessity? What's the mechanism that causes it?

I think the perspective to take would be one of 'market failures'. Either there are large improvements to be had in urban planning, or there are not. If they are not, then the whole discussion is pointless, so let's assume there are. If there are large improvements to be had, then by Coase's theorem in a functioning market, they will get made. They are not getting made. Therefore, urban planning is not a functioning 'efficient market' in any sense. Why? Rent-seeking (literally), conflicting interests, politics, expense of construction, legacy costs of infrastructure (see any major construction in NYC or Boston compared to, say, Shanghai or Beijing), etc. I don't know, ask a specialist why the dreams of urban planners so often die.

[-]billswift14y70

ask a specialist why the dreams of urban planners so often die.

I'm not a specialist, but the reason is obvious. Their dreams require other people to conform to the planners' dreams. And most peoples ideas and dreams don't. Urban planners are an extreme example of Thomas Sowell's Vision of the Anointed.

[-]michael6114y-20

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[-]khafra14y00

Heard of bizarre systems? I'm not sure that the term carves reality at the joints, or that urban planning belongs in it, but it is a model for fields that suck purely out of necessity.

[-]JoshuaZ14y50

That's an interesting link but it seems to be conflating a variety of different things. In particular, it asserts that chaotic systems need to have a lot of interacting components. This is not true. The doubling map is a very simple system that does't have much in the way of components by any reasonable definition and is chaotic. There are a lot of examples like this.

[-]khafra14y20

Thanks; if you have the time, can you point out any other structural flaws in it? All I had was the vague feeling that it wasn't as precise or rigorous as I like models to be when they claim to establish a natural type.

[-]JoshuaZ14y60

I don't know enough about some of the other fields to reliably comment but the general impression I get is that this is part of a general pattern where there are technical terms that are being used imprecisely or terms with no actual strict meaning. They seem to confuse a number of different notions of what it means by a system to have ambiguity.. While they separate the different types of ambiguity somewhat explicitly it isn't obvious to me that this is at all a helpful grouping. I don't see why it should be useful to think of the ambiguity created by self-reference in at all a similar category as the ambiguity created by incomplete information.

Also there are a handful of lines which by the most obvious notions of the terms are just wrong:

Self-references and paradoxes are much discussed problems in Philosophy and Logic alike. A Strange Loop is a situation arising in type systems where you notice that A is a kind of B which is a kind of C which is a kind of A, closing the loop.

This is not a strange loop. This is just a cycle of equivalences. All the time in mathematics one has three types of things that one wants to show are the same and one shows this by showing that A -> B, B-> C and C->A. This and variants thereof is a common proof strategy. For example, one of the most straightforward proofs that every prime that is 1 mod 4 is expressible as the sum of two perfect squares works this way.

Gödel's Theorem states that any sufficiently powerful representational system can express paradoxes

By most notions of paradoxes this statement is false. I'm not sure even if they are trying to talk about the First or Second Incompleteness theorem. But neither of them says this. Both theorems shared essence is that morally speaking, sufficiently powerful axiomatic systems with certain technical properties can talk about themselves. That's not the same claim at all. Moreover, that claim by itself isn't that surprising. If someone had just come up with Godel numbering without Godel's theorems in 1901, that probably would have been considered evidence that Hilbert's program would be successful.

But neither theorem says anywhere that sufficiently powerful systems have paradoxical results. The technical phrasings (there are a variety of phrasing but I'm going to pick some of the easier ones) are: 1) If an axiomatic system where all proofs are listable and where one can with a primitive recursive function determine if a given string is a valid proof in the system, and the system is strong enough to model Peano Arithmetic, then the system is either inconsistent or incomplete (in that there is a statement one can make in the language of the system whose truth or falseness is not provable in the system. 2) A system satisfying the above conditions can only prove its own consistency if it is actually inconsistent. (I'm deliberately glossing over here what it means for a system to be able to prove its own consistency. Here be technical details.)

The section on Emergent Properties is also confusing. They seem to at some level talk about systems where the "emergent properties" arise from an aggregate whole (such as the water example) and examples where they emerge from systems that have diverse interlocking parts. There's a common flaw of using "emergent" to mean "arises in a way we don't understand" and Eliezer and others have criticized this. They don't seem to be using that in this context since they include the example of a car. Unless they don't think people understand cars? I'm not completely sure what they mean here.

[-][anonymous]14y20

[-]Unnamed14y20

Rigor isn't the only thing that can make a book important and worth reading. When reading a book which is considered a "classic" in a field, which was written 50 years ago by a person who was not trained in that field, I would not expect a lot of rigor or use the book to assess the standard of research in that field. I'd focus more on the ideas, the implications of the ideas (e.g., where they conflict with existing practices), the style of thinking, and the value of the methods used. For instance, careful observation is an important technique in many fields for developing beliefs and models that are entangled with reality, though it is often more useful for generating hypotheses than for testing them.

[-][anonymous]14y10

My issue isn't so much with the book (it's an impressive achievement considering Jacobs' wasn't even a college graduate, much less a trained urban planner) but the fact that the field doesn't seem to have advanced passed it - that even today it's still one of the very best books on the subject.

[-]Unnamed14y30

The fact that an old not-very-technical book is still considered great doesn't necessarily mean that the field hasn't progressed, it could just mean that the book did a good job of presenting ideas that are still considered important today and that the book was important historically in the development of the field. Books which are recommended as an introduction to a field (often to non-specialists) don't need to contain the latest research results or in-depth data analysis.

[-]JamesCole14y00

The data necessary for such systematic examination is not available in some fields. I'm not sure about this field, but maybe it was one of them (back then at least)?

[-][anonymous]14y00

I'd expect the opposite to be true, actually - it's my impression that property records are very well kept, and that we have good historical data for them.

[-]JamesCole14y40

but the data for the kind of factors she's talking about (i've read the book, though it was a while ago) goes beyond what property records could provide.

[-][anonymous]14y-10

They wouldn't provide a complete picture, sure, but they'd still provide useful evidence for or against her hypothesis. For example, I'd expect it to be possible to use them to get some sort of measure of street diversity, and then compare that measure to city growth rates (or some other measure of success).

[-]JamesCole14y00

they might, though you have to be very careful in treating partial data as representative of the whole picture.

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